If your Joomla site is not equipped with a super SEO app that will strip away that very visible article ID: /34-the-great-article-title-alias, Joomla 3 does have a 'hidden' parameter to make the URL read as /the-great-article-title-alias, but it requires some effort, and is worth it. Once done, it's yours for your everlasting Joomla life.

Create A Plugin For The Job

A System plugin will serve best for the task because it is loaded globally, unless instructed otherwise.

1. Create the folder which will contain all the files and give it a relative namestrip_article_id

2. Create the primary function file within the folder and keep the name similar to the folder for simplicitystrip_article_id.php